Abstract

With use of plane waves as a basis for the band-structure calculation of a periodic assembly of highly refringent microspheres, it can be shown that resonance-mode frequencies of isolated dielectric spheres show up in the band structures. The strongly localized bands provided by the photonic-crystal analysis is compared with exact calculations made in spherical symmetry for an isolated microsphere. This comparison sheds some light on the effectiveness of the methods based on the description of mode coupling and, in particular, on the validity of tight-binding approaches of the description of photonic band structures. In addition, examining the effect of modifying the distance separating the spheres in the lattice, makes it easy to visualize the overlap between the modes of individual spheres. Thus quantitative information is provided on the geometry needed to feed energy into low-angular-momentum morphology-dependent resonances from a sharp source of the evanescent field and on the lifetime of these modes, when the resonances are disturbed by the proximity of a dielectric object of similar radius.

© 2005 Optical Society of America

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References

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  1. C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, New York, 1995).
  2. E. Lidorikis, M. M. Sigalas, E. N. Economou, C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
    [Crossref]
  3. M. Kafesaki, E. N. Economou, M. M. Sigalas, “Elastic waves in periodic composite materials,” in Photonic Band Gap Materials (Kluwer, Dordrecht, The Netherlands, 1996), p. 143 .
    [Crossref]
  4. S. Datta, C. T. Chan, K. M. Ho, C. M. Soukoulis, E. N. Economou, “Photonic band gaps in periodic dielectric structures: relation to the single-scatterer Mie resonances,” in Photonic Band Gaps and Localization (Plenum, New York, 1993), p. 289.
    [Crossref]
  5. J. P. Albert, C. Jouanin, D. Cassagne, D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
    [Crossref]
  6. J. P. Albert, C. Jouanin, D. Cassagne, D. Bertho, “Generalized Wannier function method for two-dimensional photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
    [Crossref]
  7. K. Bush, M. Frank, A. Garcia-Martin, D. Hermann, S. F. Mingaleev, M. Schillinger, L. Tkeshelashvili, “A solid state theoretical approach to the optical properties of photonic crystals,” Phys. Status Solidi A 197, 637–647 (2003).
    [Crossref]
  8. G. Mie, “Beitrage zur optik trüber medien, spellzien kolloïdaler metallosungen,” Ann. Phys. 25, 377–452 (1908).
    [Crossref]
  9. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  10. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals, Molding the Flow of Light (Princeton U. Press, Princeton N. J., 1995).
  11. J. E. Lennard-Jones, “The electronic structure of some diatomic molecules,” Trans. Faraday Soc. 25, 668–686 (1929).
    [Crossref]
  12. A. van Blaaderen, “Opals in a new light,” Science 282, 887–888 (1998).
    [Crossref]
  13. K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of photonic gaps in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [Crossref] [PubMed]
  14. S. A. Maier, P. G. Kik, H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
    [Crossref]

2003 (2)

K. Bush, M. Frank, A. Garcia-Martin, D. Hermann, S. F. Mingaleev, M. Schillinger, L. Tkeshelashvili, “A solid state theoretical approach to the optical properties of photonic crystals,” Phys. Status Solidi A 197, 637–647 (2003).
[Crossref]

S. A. Maier, P. G. Kik, H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[Crossref]

2002 (1)

J. P. Albert, C. Jouanin, D. Cassagne, D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
[Crossref]

2000 (1)

J. P. Albert, C. Jouanin, D. Cassagne, D. Bertho, “Generalized Wannier function method for two-dimensional photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
[Crossref]

1998 (2)

E. Lidorikis, M. M. Sigalas, E. N. Economou, C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[Crossref]

A. van Blaaderen, “Opals in a new light,” Science 282, 887–888 (1998).
[Crossref]

1990 (1)

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of photonic gaps in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

1929 (1)

J. E. Lennard-Jones, “The electronic structure of some diatomic molecules,” Trans. Faraday Soc. 25, 668–686 (1929).
[Crossref]

1908 (1)

G. Mie, “Beitrage zur optik trüber medien, spellzien kolloïdaler metallosungen,” Ann. Phys. 25, 377–452 (1908).
[Crossref]

Albert, J. P.

J. P. Albert, C. Jouanin, D. Cassagne, D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
[Crossref]

J. P. Albert, C. Jouanin, D. Cassagne, D. Bertho, “Generalized Wannier function method for two-dimensional photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
[Crossref]

Atwater, H. A.

S. A. Maier, P. G. Kik, H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[Crossref]

Bertho, D.

J. P. Albert, C. Jouanin, D. Cassagne, D. Bertho, “Generalized Wannier function method for two-dimensional photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
[Crossref]

Bush, K.

K. Bush, M. Frank, A. Garcia-Martin, D. Hermann, S. F. Mingaleev, M. Schillinger, L. Tkeshelashvili, “A solid state theoretical approach to the optical properties of photonic crystals,” Phys. Status Solidi A 197, 637–647 (2003).
[Crossref]

Cassagne, D.

J. P. Albert, C. Jouanin, D. Cassagne, D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
[Crossref]

J. P. Albert, C. Jouanin, D. Cassagne, D. Bertho, “Generalized Wannier function method for two-dimensional photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
[Crossref]

Chan, C. T.

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of photonic gaps in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

S. Datta, C. T. Chan, K. M. Ho, C. M. Soukoulis, E. N. Economou, “Photonic band gaps in periodic dielectric structures: relation to the single-scatterer Mie resonances,” in Photonic Band Gaps and Localization (Plenum, New York, 1993), p. 289.
[Crossref]

Datta, S.

S. Datta, C. T. Chan, K. M. Ho, C. M. Soukoulis, E. N. Economou, “Photonic band gaps in periodic dielectric structures: relation to the single-scatterer Mie resonances,” in Photonic Band Gaps and Localization (Plenum, New York, 1993), p. 289.
[Crossref]

Economou, E. N.

E. Lidorikis, M. M. Sigalas, E. N. Economou, C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[Crossref]

M. Kafesaki, E. N. Economou, M. M. Sigalas, “Elastic waves in periodic composite materials,” in Photonic Band Gap Materials (Kluwer, Dordrecht, The Netherlands, 1996), p. 143 .
[Crossref]

S. Datta, C. T. Chan, K. M. Ho, C. M. Soukoulis, E. N. Economou, “Photonic band gaps in periodic dielectric structures: relation to the single-scatterer Mie resonances,” in Photonic Band Gaps and Localization (Plenum, New York, 1993), p. 289.
[Crossref]

Frank, M.

K. Bush, M. Frank, A. Garcia-Martin, D. Hermann, S. F. Mingaleev, M. Schillinger, L. Tkeshelashvili, “A solid state theoretical approach to the optical properties of photonic crystals,” Phys. Status Solidi A 197, 637–647 (2003).
[Crossref]

Garcia-Martin, A.

K. Bush, M. Frank, A. Garcia-Martin, D. Hermann, S. F. Mingaleev, M. Schillinger, L. Tkeshelashvili, “A solid state theoretical approach to the optical properties of photonic crystals,” Phys. Status Solidi A 197, 637–647 (2003).
[Crossref]

Hermann, D.

K. Bush, M. Frank, A. Garcia-Martin, D. Hermann, S. F. Mingaleev, M. Schillinger, L. Tkeshelashvili, “A solid state theoretical approach to the optical properties of photonic crystals,” Phys. Status Solidi A 197, 637–647 (2003).
[Crossref]

Ho, K. M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of photonic gaps in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

S. Datta, C. T. Chan, K. M. Ho, C. M. Soukoulis, E. N. Economou, “Photonic band gaps in periodic dielectric structures: relation to the single-scatterer Mie resonances,” in Photonic Band Gaps and Localization (Plenum, New York, 1993), p. 289.
[Crossref]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals, Molding the Flow of Light (Princeton U. Press, Princeton N. J., 1995).

Jouanin, C.

J. P. Albert, C. Jouanin, D. Cassagne, D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
[Crossref]

J. P. Albert, C. Jouanin, D. Cassagne, D. Bertho, “Generalized Wannier function method for two-dimensional photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
[Crossref]

Kafesaki, M.

M. Kafesaki, E. N. Economou, M. M. Sigalas, “Elastic waves in periodic composite materials,” in Photonic Band Gap Materials (Kluwer, Dordrecht, The Netherlands, 1996), p. 143 .
[Crossref]

Kik, P. G.

S. A. Maier, P. G. Kik, H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[Crossref]

Kittel, C.

C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, New York, 1995).

Lennard-Jones, J. E.

J. E. Lennard-Jones, “The electronic structure of some diatomic molecules,” Trans. Faraday Soc. 25, 668–686 (1929).
[Crossref]

Lidorikis, E.

E. Lidorikis, M. M. Sigalas, E. N. Economou, C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[Crossref]

Maier, S. A.

S. A. Maier, P. G. Kik, H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals, Molding the Flow of Light (Princeton U. Press, Princeton N. J., 1995).

Mie, G.

G. Mie, “Beitrage zur optik trüber medien, spellzien kolloïdaler metallosungen,” Ann. Phys. 25, 377–452 (1908).
[Crossref]

Mingaleev, S. F.

K. Bush, M. Frank, A. Garcia-Martin, D. Hermann, S. F. Mingaleev, M. Schillinger, L. Tkeshelashvili, “A solid state theoretical approach to the optical properties of photonic crystals,” Phys. Status Solidi A 197, 637–647 (2003).
[Crossref]

Monge, D.

J. P. Albert, C. Jouanin, D. Cassagne, D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
[Crossref]

Schillinger, M.

K. Bush, M. Frank, A. Garcia-Martin, D. Hermann, S. F. Mingaleev, M. Schillinger, L. Tkeshelashvili, “A solid state theoretical approach to the optical properties of photonic crystals,” Phys. Status Solidi A 197, 637–647 (2003).
[Crossref]

Sigalas, M. M.

E. Lidorikis, M. M. Sigalas, E. N. Economou, C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[Crossref]

M. Kafesaki, E. N. Economou, M. M. Sigalas, “Elastic waves in periodic composite materials,” in Photonic Band Gap Materials (Kluwer, Dordrecht, The Netherlands, 1996), p. 143 .
[Crossref]

Soukoulis, C. M.

E. Lidorikis, M. M. Sigalas, E. N. Economou, C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[Crossref]

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of photonic gaps in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

S. Datta, C. T. Chan, K. M. Ho, C. M. Soukoulis, E. N. Economou, “Photonic band gaps in periodic dielectric structures: relation to the single-scatterer Mie resonances,” in Photonic Band Gaps and Localization (Plenum, New York, 1993), p. 289.
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Tkeshelashvili, L.

K. Bush, M. Frank, A. Garcia-Martin, D. Hermann, S. F. Mingaleev, M. Schillinger, L. Tkeshelashvili, “A solid state theoretical approach to the optical properties of photonic crystals,” Phys. Status Solidi A 197, 637–647 (2003).
[Crossref]

van Blaaderen, A.

A. van Blaaderen, “Opals in a new light,” Science 282, 887–888 (1998).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals, Molding the Flow of Light (Princeton U. Press, Princeton N. J., 1995).

Ann. Phys. (1)

G. Mie, “Beitrage zur optik trüber medien, spellzien kolloïdaler metallosungen,” Ann. Phys. 25, 377–452 (1908).
[Crossref]

Opt. Quantum Electron. (1)

J. P. Albert, C. Jouanin, D. Cassagne, D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
[Crossref]

Phys. Rev. B (2)

J. P. Albert, C. Jouanin, D. Cassagne, D. Bertho, “Generalized Wannier function method for two-dimensional photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
[Crossref]

S. A. Maier, P. G. Kik, H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[Crossref]

Phys. Rev. Lett. (2)

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of photonic gaps in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

E. Lidorikis, M. M. Sigalas, E. N. Economou, C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[Crossref]

Phys. Status Solidi A (1)

K. Bush, M. Frank, A. Garcia-Martin, D. Hermann, S. F. Mingaleev, M. Schillinger, L. Tkeshelashvili, “A solid state theoretical approach to the optical properties of photonic crystals,” Phys. Status Solidi A 197, 637–647 (2003).
[Crossref]

Science (1)

A. van Blaaderen, “Opals in a new light,” Science 282, 887–888 (1998).
[Crossref]

Trans. Faraday Soc. (1)

J. E. Lennard-Jones, “The electronic structure of some diatomic molecules,” Trans. Faraday Soc. 25, 668–686 (1929).
[Crossref]

Other (5)

C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, New York, 1995).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals, Molding the Flow of Light (Princeton U. Press, Princeton N. J., 1995).

M. Kafesaki, E. N. Economou, M. M. Sigalas, “Elastic waves in periodic composite materials,” in Photonic Band Gap Materials (Kluwer, Dordrecht, The Netherlands, 1996), p. 143 .
[Crossref]

S. Datta, C. T. Chan, K. M. Ho, C. M. Soukoulis, E. N. Economou, “Photonic band gaps in periodic dielectric structures: relation to the single-scatterer Mie resonances,” in Photonic Band Gaps and Localization (Plenum, New York, 1993), p. 289.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Photonic band structure of an assembly of spheres of radius R = 0.3535 μ m distributed according to a face-centered cubic lattice of cell parameter (a) a = 2 μ m and (b) a = 1 μ m . The refractive index of the spheres is set to 3.6. In units 2 π a , Γ = ( 0 , 0 , 0 ) , X = ( 0 , 1 , 0 ) , L = ( 1 2 , 1 2 , 1 2 ) , and U = ( 1 4 , 1 , 1 4 ) .

Fig. 2
Fig. 2

Density of modes of the face-centered cubic crystal of cell parameter a = 2 μ m . The scatterers are spheres of radius R = 0.3535 μ m and refractive index n = 3.6 . The density of modes integrates all dispersion relations in the Brillouin zone. Dotted and solid lines represent TM and TE resonance modes of the isolated sphere, respectively.

Fig. 3
Fig. 3

Position of the second band (dashed curves) and the third band (dotted curves) according to the cell parameter expressed in micrometers. The square, circle, and cross symbols correspond, respectively, to the Γ, X, and L points of the Brillouin zone. The curve associated with the Γ point (solid curve) is the same for the second and third bands. The radius of the sphere R is 0.3535 μ m . The refractive index of the scatterers is n = 3.6 .

Fig. 4
Fig. 4

Intensity of the magnetic field on the (101) face of the face-centered cubic crystal of dielectric spheres associated with (a) the second and (b) the third bands at the X point. The cell parameter a corresponds to 2 μ m , and the radius R is equivalent to 0.3535 μ m . The refractive index of the scatterers is n = 3.6 . The white circle indicates the size of a sphere.

Fig. 5
Fig. 5

As Fig. 4 except that the cell parameter a corresponds to 1 μ m .

Equations (7)

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P [ n ρ j l ( n ρ ) ] n ρ j l ( n ρ ) = [ ρ h l ( 1 ) ( ρ ) ] ρ h l ( 1 ) ( ρ ) ,
[ TE ] : P = n ,
[ TM ] : P = 1 n .
D ( ω ) = i ω i , k = ω d S k ω i , k .
a = 2 2 π 3 f 3 R .
[ TE ] : ω R 2 π c = 2 k 1 4 n ,
[ TM ] : ω R 2 π c = k 2 n ,

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